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<div  id='write'  class = 'is-mac first-line-indent'><h1><a name="泰勒公式和展开应用同阶无穷小求解" class="md-header-anchor"></a><span>泰勒公式和展开应用(同阶无穷小求解)</span></h1><blockquote><p><span>Author:张一极</span></p><p><span>个人博客:</span><a href='http://likedge.top'><span>极度空间</span></a></p><p><span>公众号:视觉迷航</span></p></blockquote><p>&nbsp;</p><h1><a name="知识" class="md-header-anchor"></a><span>知识</span></h1><blockquote><p><span>Tips:函数差值极限为0,不可以证明两个函数极限相等,因为两个函数的极限未必一定存在</span></p><p><span>不是任意的无穷小都可以比阶</span></p><p><span>两个无穷小的比值为c不为0,则分子是分母的同阶无穷小</span></p><p><span>比值为0,则分子是分母的高阶无穷小</span></p><p><span>比值为</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.323ex" height="1.503ex" viewBox="0 -538 1000 647.2" role="img" focusable="false" style="vertical-align: -0.254ex;"><defs><path stroke-width="0" id="E12-MJMAIN-221E" d="M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E12-MJMAIN-221E" x="0" y="0"></use></g></svg></span><script type="math/tex">\infty</script><span>,分子是分母的低阶无穷小</span></p><p><span>比值为1,分子是分母的等价无穷小</span></p><p><span>a(x)比上</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.511ex" height="2.872ex" viewBox="0 -913.1 2803.4 1236.7" role="img" focusable="false" style="vertical-align: 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0 0)"><use xlink:href="#E13-MJMAIN-5B" x="0" y="0"></use><use xlink:href="#E13-MJMATHI-62" x="278" y="0"></use><use xlink:href="#E13-MJMAIN-28" x="707" y="0"></use><use xlink:href="#E13-MJMATHI-78" x="1096" y="0"></use><use xlink:href="#E13-MJMAIN-29" x="1668" y="0"></use><g transform="translate(2057,0)"><use xlink:href="#E13-MJMAIN-5D" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E13-MJMATHI-6B" x="393" y="513"></use></g></g></svg></span><script type="math/tex">[b(x)]^k</script><span>=c不为0,则a是b的k阶无穷小</span></p></blockquote><p><img src="image-20200518120558882.png" referrerpolicy="no-referrer" alt="image-20200518120558882"></p><h2><a name="泰勒公式" class="md-header-anchor"></a><span>泰勒公式:</span></h2><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="73.818ex" height="16.814ex" viewBox="0 -3860.9 31782.6 7239.5" role="img" focusable="false" style="vertical-align: -7.594ex; 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xlink:href="#E26-MJMAIN-31" x="17297" y="0"></use><use xlink:href="#E26-MJMAIN-2B" x="18020" y="0"></use><use xlink:href="#E26-MJMATHI-78" x="19020" y="0"></use><g transform="translate(19592,0)"><use xlink:href="#E26-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E26-MJMATHI-61" x="550" y="513"></use></g><use xlink:href="#E26-MJMAIN-2212" x="20677" y="0"></use><use xlink:href="#E26-MJMAIN-31" x="21677" y="0"></use><use xlink:href="#E26-MJMAIN-223C" x="22455" y="0"></use><use xlink:href="#E26-MJMATHI-61" x="23511" y="0"></use><use xlink:href="#E26-MJMAIN-22C5" x="24262" y="0"></use><use xlink:href="#E26-MJMATHI-78" x="24762" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-2">\begin{array}{l}\sin x \sim x, \quad \text { tan } x \sim x, \quad \text { aresin } x \sim x, \quad \text { arctan } x \sim x, \quad \ln (1+x) \sim x, \quad \mathrm{e}^{x}-1 \sim x \\ \quad a^{\prime}-1 \sim x \ln a, \quad 1-\cos x \sim \frac{1}{2} x^{2}, \quad(1+x)^{a}-1 \sim a \cdot x\end{array}</script></div></div><h1><a name="泰勒公式展开应用" class="md-header-anchor"></a><span>泰勒公式展开应用:</span></h1><h3><a name="1fracab型" class="md-header-anchor"></a><span>1.</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.083ex" height="3.176ex" viewBox="0 -956.3 896.7 1367.2" role="img" focusable="false" style="vertical-align: -0.955ex;"><defs><path stroke-width="0" id="E15-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 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46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path stroke-width="0" id="E16-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 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190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path stroke-width="0" id="E16-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E16-MJMAIN-6C"></use><use xlink:href="#E16-MJMAIN-69" x="278" y="0"></use><use xlink:href="#E16-MJMAIN-6D" x="556" y="0"></use><g transform="translate(1389,-150)"><use transform="scale(0.707)" xlink:href="#E16-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E16-MJMAIN-2192" x="572" y="0"></use><use transform="scale(0.707)" xlink:href="#E16-MJMAIN-30" x="1572" y="0"></use></g><g transform="translate(2954,0)"><g transform="translate(286,0)"><rect stroke="none" width="2514" height="60" x="0" y="220"></rect><g transform="translate(60,468)"><use transform="scale(0.707)" xlink:href="#E16-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E16-MJMAIN-2212" x="572" y="0"></use><g transform="translate(954,0)"><use transform="scale(0.707)" xlink:href="#E16-MJMAIN-73"></use><use transform="scale(0.707)" xlink:href="#E16-MJMAIN-69" x="394" y="0"></use><use transform="scale(0.707)" xlink:href="#E16-MJMAIN-6E" x="672" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E16-MJMATHI-78" x="2813" y="0"></use></g><g transform="translate(894,-448)"><use transform="scale(0.707)" xlink:href="#E16-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E16-MJMAIN-33" x="808" y="408"></use></g></g></g></g></svg></span><script type="math/tex">\lim _{x \rightarrow 0} \frac{x-\sin x}{x^{3}}</script></p><h3><a name="解" class="md-header-anchor"></a><span>解:</span></h3><p><span>求#</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n24" cid="n24" mdtype="math_block">
			
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y="513"></use></g><use xlink:href="#E27-MJMAIN-2B" x="1961" y="0"></use><use xlink:href="#E27-MJMATHI-6F" x="2961" y="0"></use><g transform="translate(3613,0)"><use xlink:href="#E27-MJSZ1-28"></use><g transform="translate(458,0)"><use xlink:href="#E27-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E27-MJMAIN-35" x="808" y="513"></use></g><use xlink:href="#E27-MJSZ1-29" x="1483" y="-1"></use></g></g><g transform="translate(2324,-734)"><use xlink:href="#E27-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E27-MJMAIN-31" x="808" y="408"></use></g></g></g><use xlink:href="#E27-MJMAIN-2212" x="8824" y="0"></use><g transform="translate(9602,0)"><g transform="translate(342,0)"><rect stroke="none" width="620" height="60" x="0" y="220"></rect><use xlink:href="#E27-MJMAIN-31" x="60" y="676"></use><use xlink:href="#E27-MJMAIN-36" x="60" y="-686"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-3">\lim _{x \rightarrow 0} \frac{x-\sin x}{x^{3}}\\=\lim _{x \rightarrow 0} \frac{x-[x-\frac{1}{6}x^3+o(x^5)]}{x^{3}}\\=\lim _{x \rightarrow 0} \frac{\frac{1}{6} x^{3}+o\left(x^{5}\right)}{x^{1}}-\frac{1}{6}</script></div></div><hr /><h3><a name="2a-b型幂次最低原则" class="md-header-anchor"></a><span>2.</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.344ex" height="2.146ex" viewBox="0 -785.9 2731.4 924.2" role="img" focusable="false" style="vertical-align: -0.321ex;"><defs><path stroke-width="0" id="E17-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 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stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E17-MJMATHI-41" x="0" y="0"></use><use xlink:href="#E17-MJMAIN-2212" x="972" y="0"></use><use xlink:href="#E17-MJMATHI-42" x="1972" y="0"></use></g></svg></span><script type="math/tex">A-B</script><span>型,幂次最低原则:</span></h3><blockquote><p><span>应该展开到系数不相同的最低次幂</span></p></blockquote><h3><a name="问-n28" class="md-header-anchor"></a><span>问</span></h3><p><span>已知当</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.103ex" height="2.001ex" viewBox="0 -752.3 2627.6 861.5" role="img" focusable="false" style="vertical-align: -0.254ex;"><defs><path stroke-width="0" id="E18-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E18-MJMAIN-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E18-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 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386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z"></path><path stroke-width="0" id="E19-MJMATHI-6F" d="M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z"></path><path stroke-width="0" id="E19-MJMATHI-73" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path stroke-width="0" id="E19-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E19-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 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stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E19-MJMATHI-63" x="0" y="0"></use><use xlink:href="#E19-MJMATHI-6F" x="433" y="0"></use><use xlink:href="#E19-MJMATHI-73" x="918" y="0"></use><use xlink:href="#E19-MJMATHI-78" x="1387" y="0"></use><use xlink:href="#E19-MJMAIN-2212" x="2181" y="0"></use><g transform="translate(3181,0)"><use xlink:href="#E19-MJMATHI-65" x="0" y="0"></use><g transform="translate(466,424)"><use transform="scale(0.707)" xlink:href="#E19-MJMAIN-2212" x="0" y="0"></use><g transform="translate(550,0)"><g transform="translate(120,0)"><rect stroke="none" width="706" height="60" x="0" y="146"></rect><g transform="translate(60,342)"><use transform="scale(0.5)" xlink:href="#E19-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E19-MJMAIN-32" x="572" y="362"></use></g><use transform="scale(0.5)" xlink:href="#E19-MJMAIN-32" x="455" y="-634"></use></g></g></g></g></g></svg></span><script type="math/tex">cosx-e^{-\frac{x^2}{2}}</script><span>与</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.494ex" height="2.374ex" viewBox="0 -913.1 1504.3 1022.3" role="img" focusable="false" style="vertical-align: -0.254ex;"><defs><path stroke-width="0" id="E20-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="0" id="E20-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E20-MJMATHI-62" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E20-MJMATHI-61" x="0" y="0"></use><g transform="translate(529,0)"><use xlink:href="#E20-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E20-MJMATHI-62" x="808" y="513"></use></g></g></svg></span><script type="math/tex">ax^b</script><span>为等价无穷小,求a,b.</span></p><h3><a name="解-n30" class="md-header-anchor"></a><span>解:</span></h3><p><span>首先根据泰勒展开式:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n32" cid="n32" mdtype="math_block">
			
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391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E28-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E28-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 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y="0"></use><use transform="scale(0.707)" xlink:href="#E28-MJMAIN-34" x="808" y="583"></use></g><use xlink:href="#E28-MJMAIN-29" x="12733" y="0"></use></g><g transform="translate(13845,-5108)"><use xlink:href="#E28-MJMAIN-2192" x="0" y="0"></use><g transform="translate(1000,0)"><g transform="translate(397,0)"><rect stroke="none" width="1120" height="60" x="0" y="220"></rect><use xlink:href="#E28-MJMAIN-31" x="310" y="676"></use><g transform="translate(60,-686)"><use xlink:href="#E28-MJMAIN-32"></use><use xlink:href="#E28-MJMAIN-34" x="500" y="0"></use></g></g></g><g transform="translate(2637,0)"><use xlink:href="#E28-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E28-MJMAIN-34" x="808" y="583"></use></g><use xlink:href="#E28-MJMAIN-2212" x="3885" y="0"></use><g transform="translate(4663,0)"><g transform="translate(342,0)"><rect stroke="none" width="620" height="60" x="0" y="220"></rect><use xlink:href="#E28-MJMAIN-31" x="60" y="676"></use><use xlink:href="#E28-MJMAIN-38" x="60" y="-686"></use></g></g><g transform="translate(5745,0)"><use xlink:href="#E28-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E28-MJMAIN-34" x="808" y="583"></use></g></g><g transform="translate(13365,-7589)"><use xlink:href="#E28-MJMAIN-3D" x="0" y="0"></use><use xlink:href="#E28-MJMAIN-2212" x="1055" y="0"></use><g transform="translate(1833,0)"><g transform="translate(120,0)"><rect stroke="none" width="1120" height="60" x="0" y="220"></rect><use xlink:href="#E28-MJMAIN-31" x="310" y="676"></use><g transform="translate(60,-686)"><use xlink:href="#E28-MJMAIN-31"></use><use xlink:href="#E28-MJMAIN-32" x="500" y="0"></use></g></g></g><g transform="translate(3193,0)"><use xlink:href="#E28-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E28-MJMAIN-34" x="808" y="583"></use></g><use xlink:href="#E28-MJMAIN-2B" x="4441" y="0"></use><use xlink:href="#E28-MJMATHI-6F" x="5441" y="0"></use><use xlink:href="#E28-MJMAIN-28" x="5926" y="0"></use><g transform="translate(6315,0)"><use xlink:href="#E28-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E28-MJMAIN-34" x="808" y="583"></use></g><use xlink:href="#E28-MJMAIN-29" x="7341" y="0"></use></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-4">cosx=1-\frac{x^2}{2}+\frac{1}{24}x^4+o(x^4)
\\e^{-\frac{x^2}{2}} = 1-\frac{x^2}{2}+\frac{1}{8}x^4+o(x^4)
\\\to\frac{1}{24}x^4-\frac{1}{8}x^4
\\=-\frac{1}{12}x^4+o(x^4)</script></div></div><p><span>得:</span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.611ex" height="5.984ex" viewBox="0 -966.7 3707.7 2576.6" role="img" focusable="false" style="vertical-align: -3.739ex;"><defs><path stroke-width="0" id="E21-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="0" id="E21-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E21-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E21-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E21-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="0" id="E21-MJMATHI-62" d="M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z"></path><path stroke-width="0" id="E21-MJMAIN-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E21-MJMATHI-61" x="0" y="0"></use><use xlink:href="#E21-MJMAIN-3D" x="806" y="0"></use><use xlink:href="#E21-MJMAIN-2212" x="1862" y="0"></use><g transform="translate(2640,0)"><g transform="translate(120,0)"><rect stroke="none" width="827" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xlink:href="#E21-MJMAIN-31" x="334" y="580"></use><g transform="translate(60,-382)"><use transform="scale(0.707)" xlink:href="#E21-MJMAIN-31"></use><use transform="scale(0.707)" xlink:href="#E21-MJMAIN-32" x="500" y="0"></use></g></g></g><g transform="translate(0,-1506)"><use xlink:href="#E21-MJMATHI-62" x="0" y="0"></use><use xlink:href="#E21-MJMAIN-3D" x="706" y="0"></use><use xlink:href="#E21-MJMAIN-34" x="1762" y="0"></use></g></g></svg></span><script type="math/tex">a = -\frac{1}{12}\\b = 4</script></p><hr /><h3><a name="问-n92" class="md-header-anchor"></a><span>问</span></h3><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n61" cid="n61" mdtype="math_block">
			
		<div class="md-rawblock-container md-math-container" contenteditable="false" tabindex="-1">
						<div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-199-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="23.311ex" height="5.362ex" viewBox="0 -1449.1 10036.6 2308.6" role="img" focusable="false" style="vertical-align: -1.996ex; max-width: 100%;"><defs><path stroke-width="0" id="E244-MJMAIN-6C" d="M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z"></path><path stroke-width="0" id="E244-MJMAIN-69" d="M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 46H255V0H247Z"></path><path stroke-width="0" id="E244-MJMAIN-6D" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path stroke-width="0" id="E244-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E244-MJMAIN-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E244-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E244-MJMAIN-61" d="M137 305T115 305T78 320T63 359Q63 394 97 421T218 448Q291 448 336 416T396 340Q401 326 401 309T402 194V124Q402 76 407 58T428 40Q443 40 448 56T453 109V145H493V106Q492 66 490 59Q481 29 455 12T400 -6T353 12T329 54V58L327 55Q325 52 322 49T314 40T302 29T287 17T269 6T247 -2T221 -8T190 -11Q130 -11 82 20T34 107Q34 128 41 147T68 188T116 225T194 253T304 268H318V290Q318 324 312 340Q290 411 215 411Q197 411 181 410T156 406T148 403Q170 388 170 359Q170 334 154 320ZM126 106Q126 75 150 51T209 26Q247 26 276 49T315 109Q317 116 318 175Q318 233 317 233Q309 233 296 232T251 223T193 203T147 166T126 106Z"></path><path stroke-width="0" id="E244-MJMAIN-72" d="M36 46H50Q89 46 97 60V68Q97 77 97 91T98 122T98 161T98 203Q98 234 98 269T98 328L97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 60 434T96 436Q112 437 131 438T160 441T171 442H174V373Q213 441 271 441H277Q322 441 343 419T364 373Q364 352 351 337T313 322Q288 322 276 338T263 372Q263 381 265 388T270 400T273 405Q271 407 250 401Q234 393 226 386Q179 341 179 207V154Q179 141 179 127T179 101T180 81T180 66V61Q181 59 183 57T188 54T193 51T200 49T207 48T216 47T225 47T235 46T245 46H276V0H267Q249 3 140 3Q37 3 28 0H20V46H36Z"></path><path stroke-width="0" id="E244-MJMAIN-63" d="M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z"></path><path stroke-width="0" id="E244-MJMAIN-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path><path stroke-width="0" id="E244-MJMAIN-6E" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q450 438 463 329Q464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z"></path><path stroke-width="0" id="E244-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E244-MJMAIN-74" d="M27 422Q80 426 109 478T141 600V615H181V431H316V385H181V241Q182 116 182 100T189 68Q203 29 238 29Q282 29 292 100Q293 108 293 146V181H333V146V134Q333 57 291 17Q264 -10 221 -10Q187 -10 162 2T124 33T105 68T98 100Q97 107 97 248V385H18V422H27Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(38,0)"><use xlink:href="#E244-MJMAIN-6C"></use><use xlink:href="#E244-MJMAIN-69" x="278" y="0"></use><use xlink:href="#E244-MJMAIN-6D" x="556" y="0"></use></g><g transform="translate(0,-638)"><use transform="scale(0.707)" xlink:href="#E244-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E244-MJMAIN-2192" x="572" y="0"></use><use transform="scale(0.707)" xlink:href="#E244-MJMAIN-30" x="1572" y="0"></use></g><g transform="translate(1465,0)"><g transform="translate(286,0)"><rect stroke="none" width="8164" height="60" x="0" y="220"></rect><g transform="translate(60,676)"><use xlink:href="#E244-MJMAIN-61"></use><use xlink:href="#E244-MJMAIN-72" x="500" y="0"></use><use xlink:href="#E244-MJMAIN-63" x="892" y="0"></use><use xlink:href="#E244-MJMAIN-73" x="1336" y="0"></use><use xlink:href="#E244-MJMAIN-69" x="1730" y="0"></use><use xlink:href="#E244-MJMAIN-6E" x="2008" y="0"></use><use xlink:href="#E244-MJMATHI-78" x="2730" y="0"></use><use xlink:href="#E244-MJMAIN-2212" x="3524" y="0"></use><g transform="translate(4525,0)"><use xlink:href="#E244-MJMAIN-61"></use><use xlink:href="#E244-MJMAIN-72" x="500" y="0"></use><use xlink:href="#E244-MJMAIN-63" x="892" y="0"></use><use xlink:href="#E244-MJMAIN-74" x="1336" y="0"></use><use xlink:href="#E244-MJMAIN-61" x="1725" y="0"></use><use xlink:href="#E244-MJMAIN-6E" x="2225" y="0"></use></g><use xlink:href="#E244-MJMATHI-78" x="7472" y="0"></use></g><g transform="translate(1396,-686)"><use xlink:href="#E244-MJMAIN-73"></use><use xlink:href="#E244-MJMAIN-69" x="394" y="0"></use><use xlink:href="#E244-MJMAIN-6E" x="672" y="0"></use><use xlink:href="#E244-MJMATHI-78" x="1394" y="0"></use><use xlink:href="#E244-MJMAIN-2212" x="2188" y="0"></use><g transform="translate(3189,0)"><use xlink:href="#E244-MJMAIN-74"></use><use xlink:href="#E244-MJMAIN-61" x="389" y="0"></use><use xlink:href="#E244-MJMAIN-6E" x="889" y="0"></use></g><use xlink:href="#E244-MJMATHI-78" x="4800" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-199">\lim _{x \rightarrow 0} \frac{\arcsin x-\arctan x}{\sin x-\tan x}</script>
					</div></div><h3><a name="解-n59" class="md-header-anchor"></a><span>解:</span></h3><p><span>根据泰勒展开:</span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="22.553ex" height="3.619ex" viewBox="0 -1073.9 9710.2 1558.3" role="img" focusable="false" style="vertical-align: -1.125ex;"><defs><path stroke-width="0" id="E342-MJMATHI-73" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path stroke-width="0" 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transform="scale(0.707)" xlink:href="#E245-MJMAIN-61" x="1725" y="0"></use><use transform="scale(0.707)" xlink:href="#E245-MJMAIN-6E" x="2225" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E245-MJMATHI-78" x="7166" y="0"></use></g><g transform="translate(1004,-384)"><use transform="scale(0.707)" xlink:href="#E245-MJMAIN-73"></use><use transform="scale(0.707)" xlink:href="#E245-MJMAIN-69" x="394" y="0"></use><use transform="scale(0.707)" xlink:href="#E245-MJMAIN-6E" x="672" y="0"></use><use transform="scale(0.707)" xlink:href="#E245-MJMATHI-78" x="1463" y="0"></use><use transform="scale(0.707)" xlink:href="#E245-MJMAIN-2212" x="2035" y="0"></use><g transform="translate(1989,0)"><use transform="scale(0.707)" xlink:href="#E245-MJMAIN-74"></use><use transform="scale(0.707)" xlink:href="#E245-MJMAIN-61" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E245-MJMAIN-6E" x="888" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E245-MJMATHI-78" x="4494" y="0"></use></g></g></g></g></svg></span><script type="math/tex">\lim _{x \rightarrow 0} \frac{\arcsin x-\arctan x}{\sin x-\tan x}</script></p><p><span> =</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.324ex" height="6.607ex" viewBox="0 -1663.5 4445.1 2844.6" role="img" focusable="false" style="vertical-align: -2.743ex;"><defs><path stroke-width="0" id="E368-MJMATHI-6C" d="M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z"></path><path stroke-width="0" id="E368-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E368-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E368-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E368-MJMAIN-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E368-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E368-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path stroke-width="0" id="E368-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="0" id="E368-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E368-MJMATHI-6C" x="0" y="0"></use><use xlink:href="#E368-MJMATHI-69" x="298" y="0"></use><g transform="translate(643,0)"><use xlink:href="#E368-MJMATHI-6D" x="0" y="0"></use><g transform="translate(878,-150)"><use transform="scale(0.707)" xlink:href="#E368-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E368-MJMAIN-2192" x="572" y="0"></use><use transform="scale(0.707)" xlink:href="#E368-MJMAIN-30" x="1572" y="0"></use></g></g><g transform="translate(3086,0)"><g transform="translate(120,0)"><rect stroke="none" width="1119" height="60" x="0" y="220"></rect><g transform="translate(86,727)"><g transform="translate(120,0)"><rect stroke="none" width="706" height="60" x="0" y="146"></rect><g transform="translate(60,342)"><use transform="scale(0.5)" xlink:href="#E368-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E368-MJMAIN-33" x="572" y="362"></use></g><use transform="scale(0.5)" xlink:href="#E368-MJMAIN-32" x="455" y="-634"></use></g></g><g transform="translate(60,-731)"><g transform="translate(120,0)"><rect stroke="none" width="759" height="60" x="0" y="146"></rect><g transform="translate(86,342)"><use transform="scale(0.5)" xlink:href="#E368-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E368-MJMAIN-33" x="572" y="288"></use></g><g transform="translate(60,-317)"><use transform="scale(0.5)" xlink:href="#E368-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E368-MJMAIN-32" x="778" y="0"></use></g></g></g></g></g></g></svg></span><script type="math/tex">lim_{x\to0}\frac{\frac{x^3}{2}}{\frac{x^3}{-2}}</script></p><p><span> =</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.968ex" height="2.125ex" viewBox="0 -752.3 1278 915.1" role="img" focusable="false" style="vertical-align: -0.378ex;"><defs><path stroke-width="0" id="E377-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E377-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E377-MJMAIN-2212" x="0" y="0"></use><use xlink:href="#E377-MJMAIN-31" x="778" y="0"></use></g></svg></span><script type="math/tex">-1</script></p><hr /><h3><a name="问-n36" class="md-header-anchor"></a><span>问</span></h3><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="15.948ex" height="2.374ex" viewBox="0 -752.3 6866.6 1022.3" role="img" focusable="false" style="vertical-align: -0.627ex;"><defs><path stroke-width="0" id="E22-MJMATHI-6C" d="M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z"></path><path stroke-width="0" id="E22-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E22-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E22-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E22-MJMAIN-2192" d="M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E22-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E22-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E22-MJMATHI-73" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path stroke-width="0" id="E22-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E22-MJMATHI-6C" x="0" y="0"></use><use xlink:href="#E22-MJMATHI-69" x="298" y="0"></use><g transform="translate(643,0)"><use xlink:href="#E22-MJMATHI-6D" x="0" y="0"></use><g transform="translate(878,-150)"><use transform="scale(0.707)" xlink:href="#E22-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E22-MJMAIN-2192" x="572" y="0"></use><use transform="scale(0.707)" xlink:href="#E22-MJMAIN-30" x="1572" y="0"></use></g></g><use xlink:href="#E22-MJMATHI-78" x="3086" y="0"></use><use xlink:href="#E22-MJMAIN-2B" x="3880" y="0"></use><use xlink:href="#E22-MJMATHI-73" x="4880" y="0"></use><use xlink:href="#E22-MJMATHI-69" x="5349" y="0"></use><use xlink:href="#E22-MJMATHI-6E" x="5694" y="0"></use><use xlink:href="#E22-MJMATHI-78" x="6294" y="0"></use></g></svg></span><script type="math/tex">lim_{x\to0}x+sinx</script></p><h3><a name="解-n38" class="md-header-anchor"></a><span>解:</span></h3><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n39" cid="n39" mdtype="math_block">
			
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xlink:href="#E29-MJMATHI-78" x="4153" y="0"></use><use xlink:href="#E29-MJMAIN-2192" x="5003" y="0"></use><use xlink:href="#E29-MJMAIN-2212" x="6280" y="0"></use><g transform="translate(7058,0)"><g transform="translate(120,0)"><rect stroke="none" width="620" height="60" x="0" y="220"></rect><use xlink:href="#E29-MJMAIN-31" x="60" y="676"></use><use xlink:href="#E29-MJMAIN-36" x="60" y="-686"></use></g></g><g transform="translate(7918,0)"><use xlink:href="#E29-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E29-MJMAIN-33" x="808" y="583"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-5">x+sinx = x-(-sinx)
\\x = 1\cdot x
\\-sinx = -1\cdot x+\frac{1}{6}x^3+o(x)
\\x+sinx=[2x+o(x)]\to2x
\\同理(x-sinx)\to\frac{1}{6}x^3
\\\therefore tanx-x\to\frac{1}{3}x^2
\\\therefore sinx-x \to-\frac{1}{6}x^3</script></div></div><hr /><h3><a name="3等价无穷小替换" class="md-header-anchor"></a><span>3.等价无穷小替换</span></h3><h3><a name="问-n42" class="md-header-anchor"></a><span>问</span></h3><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n43" cid="n43" mdtype="math_block">
			
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y="513"></use></g></g><g transform="translate(887,-806)"><use xlink:href="#E30-MJMATHI-65" x="0" y="0"></use><g transform="translate(466,288)"><use transform="scale(0.707)" xlink:href="#E30-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E30-MJMAIN-34" x="808" y="408"></use></g><use xlink:href="#E30-MJMAIN-2212" x="1513" y="0"></use><use xlink:href="#E30-MJMAIN-31" x="2513" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-6">\lim _{x \rightarrow 0} \frac{\sin ^{2} x-x^{2}}{e^{x^{4}}-1}</script></div></div><h3><a name="解-n44" class="md-header-anchor"></a><span>解:</span></h3><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n45" cid="n45" mdtype="math_block">
			
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xlink:href="#E31-MJMATHI-78" x="3597" y="0"></use><use xlink:href="#E31-MJMAIN-29" x="4169" y="0"></use><use xlink:href="#E31-MJMAIN-28" x="4558" y="0"></use><use xlink:href="#E31-MJMATHI-73" x="4947" y="0"></use><use xlink:href="#E31-MJMATHI-69" x="5416" y="0"></use><use xlink:href="#E31-MJMATHI-6E" x="5761" y="0"></use><use xlink:href="#E31-MJMATHI-78" x="6361" y="0"></use><use xlink:href="#E31-MJMAIN-2212" x="7155" y="0"></use><use xlink:href="#E31-MJMATHI-78" x="8155" y="0"></use><use xlink:href="#E31-MJMAIN-29" x="8727" y="0"></use></g><g transform="translate(4105,-742)"><use xlink:href="#E31-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E31-MJMAIN-34" x="808" y="408"></use></g></g></g><use xlink:href="#E31-MJMAIN-2192" x="19602" y="0"></use><g transform="translate(20879,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(53.596) matrix(1 0 0 -1 0 0)">等</text></g><g transform="translate(21542,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(53.596) matrix(1 0 0 -1 0 0)">价</text></g><g transform="translate(22204,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(53.596) matrix(1 0 0 -1 0 0)">无</text></g><g transform="translate(22867,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(53.596) matrix(1 0 0 -1 0 0)">穷</text></g><g transform="translate(23529,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(53.596) matrix(1 0 0 -1 0 0)">小</text></g><g transform="translate(24191,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(53.596) matrix(1 0 0 -1 0 0)">替</text></g><g transform="translate(24854,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(53.596) matrix(1 0 0 -1 0 0)">换</text></g><g transform="translate(25516,0)"><text font-family="STIXGeneral,'Arial Unicode MS',serif" stroke="none" transform="scale(53.596) matrix(1 0 0 -1 0 0)">得</text></g><use xlink:href="#E31-MJMAIN-3A" x="26456" y="0"></use><g transform="translate(27012,0)"><g transform="translate(38,0)"><use xlink:href="#E31-MJMAIN-6C"></use><use xlink:href="#E31-MJMAIN-69" x="278" y="0"></use><use xlink:href="#E31-MJMAIN-6D" x="556" y="0"></use></g><g transform="translate(0,-638)"><use transform="scale(0.707)" xlink:href="#E31-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E31-MJMAIN-2192" x="572" y="0"></use><use transform="scale(0.707)" xlink:href="#E31-MJMAIN-30" x="1572" y="0"></use></g></g><g transform="translate(28477,0)"><g transform="translate(286,0)"><rect stroke="none" width="5265" height="60" x="0" y="220"></rect><g transform="translate(60,870)"><use xlink:href="#E31-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E31-MJMAIN-32" x="389" y="0"></use><use xlink:href="#E31-MJMATHI-78" x="889" y="0"></use><use xlink:href="#E31-MJMAIN-29" x="1461" y="0"></use><use xlink:href="#E31-MJMAIN-28" x="1850" y="0"></use><use xlink:href="#E31-MJMAIN-2212" x="2239" y="0"></use><g transform="translate(3017,0)"><g transform="translate(120,0)"><rect stroke="none" width="473" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xlink:href="#E31-MJMAIN-31" x="84" y="580"></use><use transform="scale(0.707)" xlink:href="#E31-MJMAIN-36" x="84" y="-540"></use></g></g><g transform="translate(3730,0)"><use xlink:href="#E31-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E31-MJMAIN-33" x="808" y="513"></use></g><use xlink:href="#E31-MJMAIN-29" x="4756" y="0"></use></g><g transform="translate(2119,-742)"><use xlink:href="#E31-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E31-MJMAIN-34" x="808" y="408"></use></g></g></g><use xlink:href="#E31-MJMAIN-3D" x="34427" y="0"></use><use xlink:href="#E31-MJMAIN-2212" x="35483" y="0"></use><g transform="translate(36261,0)"><g transform="translate(120,0)"><rect stroke="none" width="620" height="60" x="0" y="220"></rect><use xlink:href="#E31-MJMAIN-31" x="60" y="676"></use><use xlink:href="#E31-MJMAIN-33" x="60" y="-686"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-7">\\e^{x^4}-1\to x^4
\\sin^2x-x^2 = (sinx+x)(sinx-x)
\\\lim _{x \rightarrow 0} \frac{\sin ^{2} x-x^{2}}{e^{x^{4}}-1}\to \lim _{x \rightarrow 0} \frac{(sinx+x)(sinx-x)}{x^{4}}
\to 等价无穷小替换得:\lim _{x \rightarrow 0} \frac{(2x)(-\frac{1}{6}x^3)}{x^{4}}=-\frac{1}{3}</script></div></div><hr /><h3><a name="4归结原则" class="md-header-anchor"></a><span>4.归结原则</span></h3><h3><a name="问-n48" class="md-header-anchor"></a><span>问:</span></h3><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n49" cid="n49" mdtype="math_block">
			
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y="0"></use><use xlink:href="#E32-MJMAIN-2192" x="2433" y="0"></use><use xlink:href="#E32-MJMAIN-221E" x="3711" y="0"></use><g transform="translate(4711,0)"><use xlink:href="#E32-MJSZ3-28"></use><g transform="translate(736,0)"><use xlink:href="#E32-MJMAIN-6E"></use><use xlink:href="#E32-MJMAIN-74" x="556" y="0"></use><use xlink:href="#E32-MJMAIN-61" x="945" y="0"></use><use xlink:href="#E32-MJMAIN-6E" x="1445" y="0"></use></g><g transform="translate(2737,0)"><g transform="translate(286,0)"><rect stroke="none" width="720" height="60" x="0" y="220"></rect><use xlink:href="#E32-MJMAIN-31" x="110" y="676"></use><use xlink:href="#E32-MJMATHI-6E" x="60" y="-686"></use></g></g><use xlink:href="#E32-MJSZ3-29" x="3863" y="-1"></use><g transform="translate(4599,1176)"><use transform="scale(0.707)" xlink:href="#E32-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E32-MJMAIN-32" x="848" y="513"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-8">\lim n\to \infty  \left(\operatorname{ntan} \frac{1}{n}\right)^{n^{2}}</script></div></div><h3><a name="解-n50" class="md-header-anchor"></a><span>解:</span></h3><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n51" cid="n51" mdtype="math_block">
			
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x="1353" y="0"></use><g transform="translate(1698,0)"><use xlink:href="#E33-MJMATHI-6D" x="0" y="0"></use><g transform="translate(878,-150)"><use transform="scale(0.707)" xlink:href="#E33-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E33-MJMAIN-2192" x="572" y="0"></use><use transform="scale(0.707)" xlink:href="#E33-MJMAIN-30" x="1572" y="0"></use></g></g><g transform="translate(4141,0)"><use xlink:href="#E33-MJMATHI-65" x="0" y="0"></use><g transform="translate(466,574)"><g transform="translate(120,0)"><rect stroke="none" width="706" height="60" x="0" y="146"></rect><use transform="scale(0.5)" xlink:href="#E33-MJMAIN-31" x="455" y="674"></use><g transform="translate(60,-462)"><use transform="scale(0.5)" xlink:href="#E33-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E33-MJMAIN-32" x="572" y="288"></use></g></g><use transform="scale(0.707)" xlink:href="#E33-MJMAIN-22C5" x="1337" y="0"></use><g transform="translate(1142,0)"><g transform="translate(120,0)"><rect stroke="none" width="1316" height="60" x="0" y="146"></rect><g transform="translate(60,716)"><g transform="translate(120,0)"><rect stroke="none" width="370" height="60" x="0" y="95"></rect><use transform="scale(0.5)" xlink:href="#E33-MJMAIN-31" x="119" y="570"></use><use transform="scale(0.5)" xlink:href="#E33-MJMAIN-33" x="119" y="-736"></use></g><g transform="translate(610,0)"><use transform="scale(0.5)" xlink:href="#E33-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E33-MJMAIN-33" x="572" y="362"></use></g></g><use transform="scale(0.5)" xlink:href="#E33-MJMATHI-78" x="1029" y="-488"></use></g></g></g></g></g><g transform="translate(16115,-30133)"><use xlink:href="#E33-MJMAIN-3D" x="0" y="0"></use><g transform="translate(1055,0)"><use xlink:href="#E33-MJMATHI-65" x="0" y="0"></use><g transform="translate(466,434)"><g transform="translate(120,0)"><rect stroke="none" width="370" height="60" x="0" y="146"></rect><use transform="scale(0.5)" xlink:href="#E33-MJMAIN-31" x="119" y="674"></use><use transform="scale(0.5)" xlink:href="#E33-MJMAIN-33" x="119" y="-633"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-9">\lim x\to0 \left(\operatorname{ntan} \frac{1}{n}\right)^{n^{2}}
\\=lim_{x\to0}(\frac{tanx}{x})^\frac{1}{x^2}
\\幂指函数:lim~u^v = e^{lim(v\cdot lnu)}
\\=lim_{x\to0}(\frac{tanx}{x})^\frac{1}{x^2}
\\=e^{lim_{x\to0}\frac{1}{x^2}ln\frac{tanx}{x}}
\\拆解:lim_{x\to0}ln\frac{tanx}{x}\\
= lim_{x\to0}ln({~1+\frac{tanx}{x}-1})\to lim_{x\to0}(\frac{tanx}{x}-1)
\\其中(\frac{tanx}{x}-1)\to0
\\=lim_{x\to0}(\frac{tanx-x}{x})
\\\because tanx-x\to\frac{1}{3}x^3
\\=lim_{x\to0}(\frac{\frac{1}{3}x^3}{x})
\\\therefore lim_{x\to0}e^{lim_{x\to0}\frac{1}{x^2}ln\frac{tanx}{x}}
\\=lim_{x\to0}e^{\frac{1}{x^2}\cdot {\frac{\frac{1}{3}x^3}{x}}}
\\=e^{\frac{1}{3}}</script></div></div><p><span>由归结原则,取</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.312ex" height="3.246ex" viewBox="0 -966.7 2717.8 1397.5" role="img" focusable="false" style="vertical-align: -1ex;"><defs><path stroke-width="0" id="E23-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E23-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E23-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E23-MJMATHI-6E" x="0" y="0"></use><use xlink:href="#E23-MJMAIN-3D" x="877" y="0"></use><g transform="translate(1655,0)"><g transform="translate(397,0)"><rect stroke="none" width="544" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xlink:href="#E23-MJMAIN-31" x="134" y="580"></use><use transform="scale(0.707)" xlink:href="#E23-MJMATHI-6E" x="84" y="-488"></use></g></g></g></svg></span><script type="math/tex">n=\frac{1}{n}</script><span>,得原式 = </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.731ex" height="2.997ex" viewBox="0 -1181.1 1176 1290.3" role="img" focusable="false" style="vertical-align: -0.254ex;"><defs><path stroke-width="0" id="E24-MJMATHI-65" d="M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z"></path><path stroke-width="0" id="E24-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E24-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E24-MJMATHI-65" x="0" y="0"></use><g transform="translate(466,434)"><g transform="translate(120,0)"><rect stroke="none" width="370" height="60" x="0" y="146"></rect><use transform="scale(0.5)" xlink:href="#E24-MJMAIN-31" x="119" y="674"></use><use transform="scale(0.5)" xlink:href="#E24-MJMAIN-33" x="119" y="-633"></use></g></g></g></svg></span><script type="math/tex">e^\frac{1}{3}</script><span>.</span></p><p>&nbsp;</p></div>
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